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Theorem dchrisum0lem1 27447

Description: Lemma for dchrisum0 27451. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)
dchrisum0lem1.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
dchrisum0.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrisum0.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrisum0.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))

**Assertion**
Ref	Expression
dchrisum0lem1	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑚,𝑦, 1 𝑚,𝑑,𝑥,𝑦,𝐶 𝐹,𝑑,𝑥,𝑦 𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝑁,𝑥,𝑦 𝜑,𝑑,𝑚,𝑥 𝑆,𝑑,𝑚,𝑥,𝑦 𝑥,𝑊 𝑚,𝑍,𝑥,𝑦 𝐷,𝑚,𝑥,𝑦 𝐿,𝑎,𝑑,𝑚,𝑥,𝑦 𝑋,𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎,𝑑) 𝑆(𝑎) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑚,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑊(𝑦,𝑚,𝑎,𝑑) 𝑍(𝑎,𝑑)

**Proof of Theorem dchrisum0lem1**
Step	Hyp	Ref	Expression
1		fzfid 13872	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		fzfid 13872	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∈ Fin)
3		fzfid 13872	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
4		elfznn 13445	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
5		elfzuz 13412	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
6	4, 5	anim12i 613	. . . . . 6 ⊢ ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
8		elfzuz 13412	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
9		elfznn 13445	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
10	8, 9	anim12ci 614	. . . . . 6 ⊢ ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
11	10	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
12		eluzelz 12734	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → 𝑚 ∈ ℤ)
13	12	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℤ)
14	13	zred 12569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ)
15		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
16		2z 12496	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
17		rpexpcl 13979	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
18	15, 16, 17	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
19	18	rpred 12926	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ)
20	19	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ)
21		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℕ)
22	21	nnrpd 12924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ⁺)
23	14, 20, 22	lemuldivd 12975	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
24	21	nnred 12132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
25	15	rprege0d 12933	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
26		flge0nn0 13716	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
27		nn0p1nn 12412	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
28	25, 26, 27	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℕ)
30		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
31		eluznn 12808	. . . . . . . . . . . 12 ⊢ ((((⌊‘𝑥) + 1) ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
32	29, 30, 31	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℕ)
33	32	nnrpd 12924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ⁺)
34	24, 20, 33	lemuldiv2d 12976	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
35	23, 34	bitr3d 281	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
36		rpcn 12893	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
37	36	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
38	37	sqvald 14042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) = (𝑥 · 𝑥))
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) = (𝑥 · 𝑥))
40		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ⁺)
41	40	rpred 12926	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ)
42		reflcl 13692	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
43		peano2re 11278	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘𝑥) ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
44	41, 42, 43	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℝ)
45		fllep1 13697	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
46	41, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ ((⌊‘𝑥) + 1))
47		eluzle 12737	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → ((⌊‘𝑥) + 1) ≤ 𝑚)
48	47	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ≤ 𝑚)
49	41, 44, 14, 46, 48	letrd 11262	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ 𝑚)
50	41, 14, 40	lemul1d 12969	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 ≤ 𝑚 ↔ (𝑥 · 𝑥) ≤ (𝑚 · 𝑥)))
51	49, 50	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 · 𝑥) ≤ (𝑚 · 𝑥))
52	39, 51	eqbrtrd 5111	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ≤ (𝑚 · 𝑥))
53	20, 41, 33	ledivmuld 12979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (((𝑥↑2) / 𝑚) ≤ 𝑥 ↔ (𝑥↑2) ≤ (𝑚 · 𝑥)))
54	52, 53	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ≤ 𝑥)
55		nnre 12124	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
56	55	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
57	20, 32	nndivred 12171	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ∈ ℝ)
58		letr 11199	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℝ ∧ ((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
59	56, 57, 41, 58	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
60	54, 59	mpan2d 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑑 ≤ 𝑥))
61	35, 60	sylbid 240	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑑 ≤ 𝑥))
62	61	pm4.71rd 562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
63		nnge1 12145	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 1 ≤ 𝑑)
64	63	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 1 ≤ 𝑑)
65		1re 11104	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
66		0lt1 11631	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
67	65, 66	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ ∧ 0 < 1)
68	22	rpregt0d 12932	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ ℝ ∧ 0 < 𝑑))
69	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ⁺)
70	69	rpregt0d 12932	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2)))
71		lediv2 12004	. . . . . . . . . . . . . 14 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑑 ∈ ℝ ∧ 0 < 𝑑) ∧ ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
72	67, 68, 70, 71	mp3an2i 1468	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
73	64, 72	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1))
74	20	recnd 11132	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℂ)
75	74	div1d 11881	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 1) = (𝑥↑2))
76	73, 75	breqtrd 5115	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ (𝑥↑2))
77		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑑 ∈ ℕ)
78		nndivre 12158	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((𝑥↑2) / 𝑑) ∈ ℝ)
79	19, 77, 78	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ∈ ℝ)
80		letr 11199	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℝ ∧ ((𝑥↑2) / 𝑑) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
81	14, 79, 20, 80	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
82	76, 81	mpan2d 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑚 ≤ (𝑥↑2)))
83	35, 82	sylbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑚 ≤ (𝑥↑2)))
84	83	pm4.71rd 562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
85	35, 62, 84	3bitr3d 309	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑)) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
86		fznnfl 13758	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
87	86	baibd 539	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
88	41, 21, 87	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
89	79	flcld 13694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ)
90		elfz5 13408	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
91	30, 89, 90	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
92		flge 13701	. . . . . . . . . 10 ⊢ ((((𝑥↑2) / 𝑑) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
93	79, 13, 92	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
94	91, 93	bitr4d 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
95	88, 94	anbi12d 632	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
96	20	flcld 13694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘(𝑥↑2)) ∈ ℤ)
97		elfz5 13408	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘(𝑥↑2)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
98	30, 96, 97	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
99		flge 13701	. . . . . . . . . 10 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
100	20, 13, 99	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
101	98, 100	bitr4d 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (𝑥↑2)))
102		fznnfl 13758	. . . . . . . . . 10 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
103	102	baibd 539	. . . . . . . . 9 ⊢ ((((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
104	57, 21, 103	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
105	101, 104	anbi12d 632	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
106	85, 95, 105	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
107	106	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))))))
108	7, 11, 107	pm5.21ndd 379	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
109		ssun2 4127	. . . . . . . 8 ⊢ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
110	28	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ ℕ)
111		nnuz 12767	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
112	110, 111	eleqtrdi 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
113		dchrisum0lem1a 27417	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))))
114	113	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥)))
115		fzsplit2 13441	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
116	112, 114, 115	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
117	109, 116	sseqtrrid 3976	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
118	117	sselda 3932	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
119		rpvmasum2.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
120		rpvmasum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
121		rpvmasum2.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
122		rpvmasum.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍)
123		rpvmasum2.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
124	123	ssrab3 4030	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
125		dchrisum0.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
126	124, 125	sselid 3930	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
127	126	eldifad 3912	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
128	127	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑋 ∈ 𝐷)
129		elfzelz 13416	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℤ)
130	129	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℤ)
131	119, 120, 121, 122, 128, 130	dchrzrhcl 27176	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
132		elfznn 13445	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℕ)
133	132	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℕ)
134	133	nnrpd 12924	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℝ⁺)
135	134	rpsqrtcld 15311	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℝ⁺)
136	135	rpcnd 12928	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℂ)
137	135	rpne0d 12931	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ≠ 0)
138	131, 136, 137	divcld 11889	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
139	4	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
140	139	nnrpd 12924	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
141	140	rpsqrtcld 15311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℝ⁺)
142	141	rpcnne0d 12935	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
143	142	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
144	143	simpld 494	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ∈ ℂ)
145	143	simprd 495	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ≠ 0)
146	138, 144, 145	divcld 11889	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
147	118, 146	syldan 591	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
148	147	anasss 466	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
149	1, 2, 3, 108, 148	fsumcom2 15673	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
150	149	mpteq2dva 5182	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
151	65	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℝ)
152		2cn 12192	. . . . . . . 8 ⊢ 2 ∈ ℂ
153	15	rpsqrtcld 15311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
154	153	rpcnd 12928	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
155		mulcl 11082	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (√‘𝑥) ∈ ℂ) → (2 · (√‘𝑥)) ∈ ℂ)
156	152, 154, 155	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
157	141	rprecred 12937	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℝ)
158	1, 157	fsumrecl 15633	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℝ)
159	158	recnd 11132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℂ)
160	159, 156	subcld 11464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) ∈ ℂ)
161		2re 12191	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
162		dchrisum0.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
163		elrege0 13346	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
164	162, 163	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
165	164	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
166		remulcl 11083	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (2 · 𝐶) ∈ ℝ)
167	161, 165, 166	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
168	167	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℝ)
169	168, 153	rerpdivcld 12957	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
170	169	recnd 11132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
171	156, 160, 170	adddird 11129	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
172	156, 159	pncan3d 11467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)))
173	172	oveq1d 7356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
174		2cnd 12195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
175	174, 154, 170	mulassd 11127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))))
176	168	recnd 11132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℂ)
177	153	rpne0d 12931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
178	176, 154, 177	divcan2d 11891	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥))) = (2 · 𝐶))
179	178	oveq2d 7357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))) = (2 · (2 · 𝐶)))
180	175, 179	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · (2 · 𝐶)))
181	180	oveq1d 7356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
182	171, 173, 181	3eqtr3d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
183	182	mpteq2dva 5182	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))))
184		remulcl 11083	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (2 · 𝐶) ∈ ℝ) → (2 · (2 · 𝐶)) ∈ ℝ)
185	161, 167, 184	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℝ)
186	185	recnd 11132	. . . . . 6 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℂ)
187	186	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (2 · 𝐶)) ∈ ℂ)
188	160, 170	mulcld 11124	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
189		rpssre 12890	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
190		o1const 15519	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · (2 · 𝐶)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
191	189, 186, 190	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
192		eqid 2730	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))))
193	192	divsqrsum 26912	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟
194		rlimdmo1 15517	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
195	193, 194	mp1i 13	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
196	176, 154, 177	divrecd 11892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) = ((2 · 𝐶) · (1 / (√‘𝑥))))
197	196	mpteq2dva 5182	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))))
198	153	rprecred 12937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (√‘𝑥)) ∈ ℝ)
199	167	recnd 11132	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℂ)
200		rlimconst 15443	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · 𝐶) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
201	189, 199, 200	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
202		sqrtlim 26903	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0
203	202	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0)
204	168, 198, 201, 203	rlimmul 15544	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))) ⇝_𝑟 ((2 · 𝐶) · 0))
205	197, 204	eqbrtrd 5111	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0))
206		rlimo1 15516	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0) → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
207	205, 206	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
208	160, 170, 195, 207	o1mul2 15524	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
209	187, 188, 191, 208	o1add2 15523	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))) ∈ 𝑂(1))
210	183, 209	eqeltrd 2829	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
211	158, 169	remulcld 11134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
212	3, 147	fsumcl 15632	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
213	1, 212	fsumcl 15632	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
214	213	abscld 15338	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
215	211	recnd 11132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
216	215	abscld 15338	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ ℝ)
217	212	abscld 15338	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
218	1, 217	fsumrecl 15633	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
219	1, 212	fsumabs 15700	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
220	169	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
221	157, 220	remulcld 11134	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
222	118, 138	syldan 591	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
223	3, 222	fsumcl 15632	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
224	223	abscld 15338	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
225		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
226		rpvmasum2.1	. . . . . . . . . . 11 ⊢ 1 = (0_g‘𝐺)
227		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
228		dchrisum0.s	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
229		dchrisum0.1	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
230	120, 122, 225, 119, 121, 226, 123, 125, 227, 162, 228, 229	dchrisum0lem1b 27446	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))
231	224, 220, 141, 230	lediv1dd 12984	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) ≤ (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)))
232	141	rpcnd 12928	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℂ)
233	141	rpne0d 12931	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ≠ 0)
234	223, 232, 233	absdivd 15357	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))))
235	3, 232, 222, 233	fsumdivc 15685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
236	235	fveq2d 6821	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
237	141	rprege0d 12933	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)))
238		absid 15195	. . . . . . . . . . . 12 ⊢ (((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)) → (abs‘(√‘𝑑)) = (√‘𝑑))
239	237, 238	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(√‘𝑑)) = (√‘𝑑))
240	239	oveq2d 7357	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)))
241	234, 236, 240	3eqtr3rd 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
242	170	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
243	242, 232, 233	divrec2d 11893	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)) = ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
244	231, 241, 243	3brtr3d 5120	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
245	1, 217, 221, 244	fsumle 15698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
246	157	recnd 11132	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℂ)
247	1, 170, 246	fsummulc1 15684	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
248	245, 247	breqtrrd 5117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
249	214, 218, 211, 219, 248	letrd 11262	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
250	211	leabsd 15314	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
251	214, 211, 216, 249, 250	letrd 11262	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
252	251	adantrr 717	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
253	151, 210, 211, 213, 252	o1le 15552	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
254	150, 253	eqeltrrd 2830	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 {crab 3393 ∖ cdif 3897 ∪ cun 3898 ⊆ wss 3900 {csn 4574 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6477 (class class class)co 7341 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 · cmul 11003 +∞cpnf 11135 < clt 11138 ≤ cle 11139 − cmin 11336 / cdiv 11766 ℕcn 12117 2c2 12172 ℕ₀cn0 12373 ℤcz 12460 ℤ_≥cuz 12724 ℝ⁺crp 12882 [,)cico 13239 ...cfz 13399 ⌊cfl 13686 seqcseq 13900 ↑cexp 13960 √csqrt 15132 abscabs 15133 ⇝ cli 15383 ⇝_𝑟 crli 15384 𝑂(1)co1 15385 Σcsu 15585 Basecbs 17112 0_gc0g 17335 ℤRHomczrh 21429 ℤ/nℤczn 21432 DChrcdchr 27163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-ec 8619 df-qs 8623 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-o1 15389 df-lo1 15390 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-qus 17405 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-nsg 19029 df-eqg 19030 df-ghm 19118 df-cntz 19222 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-rhm 20383 df-subrng 20454 df-subrg 20478 df-lmod 20788 df-lss 20858 df-lsp 20898 df-sra 21100 df-rgmod 21101 df-lidl 21138 df-rsp 21139 df-2idl 21180 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-zring 21377 df-zrh 21433 df-zn 21436 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-cmp 23295 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485 df-cxp 26486 df-dchr 27164

This theorem is referenced by: dchrisum0lem3 27450

W3C validator